https://file.aiquickdraw.com/mathbot/file/1726388656700tebuxt90.jpg

The image contains a set of mathematical expressions under the heading "Rút gọn. Viết kết quả với số mũ dương," which translates to "Simplify. Write the result with positive exponents." Let's break down and simplify each expression step by step:

### a. \( b^{-5} \times b^4 \)

Use the property of exponents: \( a^m \times a^n = a^{m+n} \).

\[
b^{-5} \times b^4 = b^{-5+4} = b^{-1} = \frac{1}{b}
\]

### b. \( 2x^{-3} \times 3x^3 \)

Combine the constants and use the property of exponents.

\[
2 \times 3 \times x^{-3+3} = 6x^0 = 6 \times 1 = 6
\]

### c. \( 4s^8 + 12s^7 \)

This is not an exponent operation but a simplification of terms by factoring.

\[
4s^8 + 12s^7 = 4s^7(s + 3)
\]

### d. \( \frac{h^{-7}}{h^4} \)

Use the quotient rule: \( \frac{a^m}{a^n} = a^{m-n} \).

\[
\frac{h^{-7}}{h^4} = h^{-7-4} = h^{-11} = \frac{1}{h^{11}}
\]

### e. \( (2x^{-3})^{-3} \)

Use the power of a power rule: \( (a^m)^n = a^{m \times n} \).

\[
(2x^{-3})^{-3} = 2^{-3}x^{9} = \frac{x^9}{8}
\]

### f. \( (c^{-2})^3 \)

Again, use the power of a power rule.

\[
(c^{-2})^3 = c^{-6} = \frac{1}{c^6}
\]

### g. \( x^{-3} + x^{-4} \)

This is an addition of terms with negative exponents.

\[
x^{-3} + x^{-4} = \frac{1}{x^3} + \frac{1}{x^4}
\]

To combine them, find a common denominator:
\[
= \frac{x}{x^4} + \frac{1}{x^4} = \frac{x+1}{x^4}
\]

### h. \( \frac{x^{-2}}{x^3} \)

Use the quotient rule.

\[
\frac{x^{-2}}{x^3} = x^{-2-3} = x^{-5} = \frac{1}{x^5}
\]

### i. \( a^4b^{-3} \times a^3b^{-2} \)

Combine the exponents for both \(a\) and \(b\).

\[
a^{4+3}b^{-3-2} = a^7b^{-5} = \frac{a^7}{b^5}
\]

### j. \( (x^3y^{-2})^3 \times (xy^3)^{-2} \)

First, apply the power of a power rule to both terms, then multiply.

\[
(x^3y^{-2})^3 = x^{9}y^{-6}, \quad (xy^3)^{-2} = x^{-2}y^{-6}
\]
\[
x^9y^{-6} \times x^{-2}y^{-6} = x^{9-2}y^{-6-6} = x^7y^{-12} = \frac{x^7}{y^{12}}
\]

### k. \( \frac{2x^5y^3}{x^2y^{-4}} \times \frac{x}{x^7} \)

Simplify each part.

\[
\frac{2x^5y^3}{x^2y^{-4}} = 2x^{5-2}y^{3-(-4)} = 2x^3y^7
\]
\[
\frac{x}{x^7} = x^{1-7} = x^{-6}
\]
Now multiply both:
\[
2x^3y^7 \times x^{-6} = 2x^{3-6}y^7 = 2x^{-3}y^7 = \frac{2y^7}{x^3}
\]

### l. \( \frac{m^3n^{-6}}{m^{-4}n^7} \times \frac{m^5n^{-9}}{m^{-2}n^4} \)

Simplify each fraction separately.

\[
\frac{m^3n^{-6}}{m^{-4}n^7} = m^{3-(-4)}n^{-6-7} = m^7n^{-13}
\]
\[
\frac{m^5n^{-9}}{m^{-2}n^4} = m^{5-(-2)}n^{-9-4} = m^7n^{-13}
\]
Now multiply both:
\[
m^7n^{-13} \times m^7n^{-13} = m^{7+7}n^{-13-13} = m^{14}n^{-26} = \frac{m^{14}}{n^{26}}
\]

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If you'd like further explanations or have any questions, feel free to ask! Here are five related questions to expand on this topic:

1. How do you simplify expressions with both negative and fractional exponents?
2. Can you explain how to solve exponential equations with different bases?
3. What are the properties of radicals, and how do they relate to exponents?
4. How can we simplify expressions involving multiple variables and exponents?
5. How do exponents behave when raised to a fraction or negative exponent?

**Tip:** Always apply exponent rules systematically, one step at a time, to avoid confusion.

Simplify the given expressions and write the result with positive exponents.

This page provides a detailed solution to a set of algebraic expressions involving exponents. Learn how to apply product, quotient, and power rules to simplify and write the result with positive exponents. Suitable for students in Grades 8-10.

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